SOLUTION: 3x-5y=15 2x-y=-4 substitution 2 variables

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Question 107679: 3x-5y=15
2x-y=-4
substitution 2 variables
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax-5%2Ay=15
2%2Ax-1%2Ay=-4

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-5%2Ay=15-3%2AxSubtract 3%2Ax from both sides

y=%2815-3%2Ax%29%2F-5 Divide both sides by -5.


Which breaks down and reduces to



y=-3%2B%283%2F5%29%2Ax Now we've fully isolated y

Since y equals -3%2B%283%2F5%29%2Ax we can substitute the expression -3%2B%283%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B-1%2Ahighlight%28%28-3%2B%283%2F5%29%2Ax%29%29=-4 Replace y with -3%2B%283%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax-1%2A%28-3%29-1%283%2F5%29x=-4 Distribute -1 to -3%2B%283%2F5%29%2Ax

2%2Ax%2B3-%283%2F5%29%2Ax=-4 Multiply



2%2Ax%2B3-%283%2F5%29%2Ax=-4 Reduce any fractions

2%2Ax-%283%2F5%29%2Ax=-4-3 Subtract 3 from both sides


2%2Ax-%283%2F5%29%2Ax=-7 Combine the terms on the right side



%2810%2F5%29%2Ax-%283%2F5%29x=-7 Make 2 into a fraction with a denominator of 5

%287%2F5%29%2Ax=-7 Now combine the terms on the left side.


cross%28%285%2F7%29%287%2F5%29%29x=%28-7%2F1%29%285%2F7%29 Multiply both sides by 5%2F7. This will cancel out 7%2F5 and isolate x

So when we multiply -7%2F1 and 5%2F7 (and simplify) we get



x=-5 <---------------------------------One answer

Now that we know that x=-5, lets substitute that in for x to solve for y

2%28-5%29-1%2Ay=-4 Plug in x=-5 into the 2nd equation

-10-1%2Ay=-4 Multiply

-1%2Ay=-4%2B10Add 10 to both sides

-1%2Ay=6 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%286%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=6%2F-1 Multiply the terms on the right side


y=-6 Reduce


So this is the other answer


y=-6<---------------------------------Other answer


So our solution is

x=-5 and y=-6

which can also look like

(-5,-6)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax-5%2Ay=15
2%2Ax-1%2Ay=-4

we get


graph of 3%2Ax-5%2Ay=15 (red) and 2%2Ax-1%2Ay=-4 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-5,-6). This verifies our answer.


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Check:

Plug in (-5,-6) into the system of equations


Let x=-5 and y=-6. Now plug those values into the equation 3%2Ax-5%2Ay=15

3%2A%28-5%29-5%2A%28-6%29=15 Plug in x=-5 and y=-6


-15%2B30=15 Multiply


15=15 Add


15=15 Reduce. Since this equation is true the solution works.


So the solution (-5,-6) satisfies 3%2Ax-5%2Ay=15



Let x=-5 and y=-6. Now plug those values into the equation 2%2Ax-1%2Ay=-4

2%2A%28-5%29-1%2A%28-6%29=-4 Plug in x=-5 and y=-6


-10%2B6=-4 Multiply


-4=-4 Add


-4=-4 Reduce. Since this equation is true the solution works.


So the solution (-5,-6) satisfies 2%2Ax-1%2Ay=-4


Since the solution (-5,-6) satisfies the system of equations


3%2Ax-5%2Ay=15
2%2Ax-1%2Ay=-4


this verifies our answer.